Stability and Tipping

Learning Objectives

  • Locate the base reaction from force and moment equilibrium.
  • Compute eccentricity and interpret the middle-third or kern region.
  • Distinguish stable equilibrium, partial contact, uplift, sliding, and tipping.
  • Compare overturning and restoring moments.
  • Apply equilibrium models to platforms, retaining walls, and cranes without implying design-code compliance.
Base resultant, kern, and tipping edgeThe base reaction shifts to satisfy moment equilibrium. The middle-third boundary, contact edge, sliding resistance, and tipping condition are distinct stability checks.R at eccentricity eB/6Htipping
Base resultant, kern, and tipping edge
The base reaction shifts to satisfy moment equilibrium. The middle-third boundary, contact edge, sliding resistance, and tipping condition are distinct stability checks.

Base-Reaction Eccentricity

Eccentricity is the distance between the resultant base reaction and the geometric center of the supporting base.

Reaction Location and Eccentricity

Base-reaction location produced by the net moment and total vertical force.

xR=Mnet∑Ve=∣xR∣x_R=\frac{M_{\mathrm{net}}}{\sum V} \qquad e=|x_R|

Variables

SymbolDescriptionUnit
xRx_RSigned base-reaction location from the base centerm
eeMagnitude of reaction eccentricity, ∣xR∣|x_R|m
MnetM_{\mathrm{net}}Net moment about the base centerkN·m
∑V\sum VTotal compressive vertical forcekN

Middle-Third Criterion

Full compression over a rectangular base under a linear pressure distribution.

∣e∣≤B6|e|\leq\frac{B}{6}

Variables

SymbolDescriptionUnit
BBBase width in the direction of eccentricitym

Equilibrium Versus Design Compliance

These simulations illustrate statics and idealized contact. They do not verify soil bearing, material strength, structural detailing, load combinations, or compliance with any design standard.

Loss of Contact

A contact reaction cannot be tensile. When the idealized resultant leaves the base, the assumed full-contact reaction is impossible and uplift or tipping must be considered.

Worked Example Summary

A 300 kN300\ \text{kN} block with a 3.0 m3.0\ \text{m} base is subjected to a 60 kN60\ \text{kN} horizontal force at 2.0 m2.0\ \text{m}. The overturning moment is 120 kN⋅m120\ \text{kN}\cdot\text{m} and e=120/300=0.40 me=120/300=0.40\ \text{m}. Since B/6=0.50 mB/6=0.50\ \text{m}, the resultant remains within the middle third in this idealized model.

Simulation 1 Instructions

Move the horizontal load and change the block geometry. Compare reaction location, sliding threshold, and tipping threshold.

Stability and Base-Resultant Suite

Concept and model scope

Track sliding, signed base-resultant migration, full contact, partial contact, and the physical tipping edge.

Simulation purpose: Move real loads and geometry while the signed moment balance, compression-only contact state, and reaction line update together.

Model scope: Rigid-base statics model with compression-only support contact. Middle-third exit indicates partial contact; physical base-edge exit is a distinct loss-of-equilibrium limit.

Verification: The reaction line is drawn only while it lies on the physical base. Signed load offsets, H/3 for the retaining-wall resultant, and the displayed compression zone use the same model coordinates as the calculations.

Centered weight

Centered weight

Centered downward weight through the base center. It contributes stabilizing moment about either edge.

300 kN
Base width

Base width

Physical support width B. Kern boundaries are ±B/6 and the physical edges are ±B/2.

3.0 m
Horizontal load

Horizontal load

Horizontal force that creates a positive overturning moment and shifts the base resultant toward the positive edge.

60 kN
Load height

Load height

Vertical distance from the base to the horizontal force line of action.

2.0 m
Static friction coefficient

Static friction coefficient

Defines the limiting base sliding resistance μsΣV. The actual static friction is only the horizontal demand while equilibrium remains admissible.

0.45
RW 300 kNH 60 kNFbaseB = 3 mkern ±0.50 m · edges ±1.50 m
full compression contact
Reaction location xR
+0.400 m

Reaction location xR

Signed from the base center. Positive is toward the positive overturning edge.

Eccentricity |e|
0.400 m

Eccentricity |e|

Kern = 0.500 m; edge = 1.500 m.

Compression width
3.000 m

Compression width

Full base width is active inside the kern. After kern exit, a compression-only triangular contact zone is used; no tensile base pressure is reported.

Contact state
full-contact

Contact state

Kern exit and physical base-edge exit are intentionally separate states.

Sliding state
static

Sliding state

Static friction supplies the demand only while demand ≤ μsΣV.

Sliding factor
2.250

Sliding factor

Limiting friction capacity divided by horizontal demand.

Stabilizing moment
450 kN·m
Overturning moment
120 kN·m
Overturning factor
3.750

Overturning factor

Restoring moment divided by overturning moment about the currently governing edge.

xR=∑MC∑V,∣e∣≤B/6;(full contact),∣e∣=B/2;(tipping)x_R=\frac{\sum M_C}{\sum V},\quad |e|\le B/6\\;\text{(full contact)},\quad |e|=B/2\\;\text{(tipping)}

Equation concept

For B/6 < |e| < B/2, the model uses compression-only partial contact. A resultant beyond B/2 is not drawn as a physical reaction because the assumed base equilibrium has been lost.

Simulation 1 Concept Question

Why does lowering the force application point improve tipping stability without changing the sliding threshold?

Simulation 2 Instructions

Shift a vertical platform load through and beyond the support footprint. Observe the combined resultant, the middle-third boundary, compression-only partial contact, impending tipping when the combined resultant reaches the support edge, and loss of the assumed base reaction only after the resultant passes that edge.

Stability and Base-Resultant Suite

Concept and model scope

Move a load through and beyond the support footprint and reject equilibrium once the combined resultant leaves the base.

Simulation purpose: Move real loads and geometry while the signed moment balance, compression-only contact state, and reaction line update together.

Model scope: Rigid-base statics model with compression-only support contact. Middle-third exit indicates partial contact; physical base-edge exit is a distinct loss-of-equilibrium limit.

Verification: The reaction line is drawn only while it lies on the physical base. Signed load offsets, H/3 for the retaining-wall resultant, and the displayed compression zone use the same model coordinates as the calculations.

Centered weight

Centered weight

Centered downward weight through the base center. It contributes stabilizing moment about either edge.

300 kN
Base width

Base width

Physical support width B. Kern boundaries are ±B/6 and the physical edges are ±B/2.

3.0 m
Moving vertical load

Moving vertical load

Downward moving platform load. Its actual signed offset contributes directly to the moment balance.

100 kN
Load offset

Load offset

Signed horizontal position from the support center. The load may move beyond the footprint; stability is then decided by the combined resultant, not by a fake clamped reaction.

0.6 m
RW 300 kNP 100 kNB = 3 mkern ±0.50 m · edges ±1.50 m
full compression contact
Reaction location xR
+0.150 m

Reaction location xR

Signed from the base center. Positive is toward the positive overturning edge.

Eccentricity |e|
0.150 m

Eccentricity |e|

Kern = 0.500 m; edge = 1.500 m.

Compression width
3.000 m

Compression width

Full base width is active inside the kern. After kern exit, a compression-only triangular contact zone is used; no tensile base pressure is reported.

Contact state
full-contact

Contact state

Kern exit and physical base-edge exit are intentionally separate states.

Sliding state
not-applicable

Sliding state

No horizontal sliding demand exists in this scenario.

Sliding factor
Not applicable

Sliding factor

Limiting friction capacity divided by horizontal demand.

Stabilizing moment
540 kN·m
Overturning moment
0 kN·m
Overturning factor
No overturning demand

Overturning factor

Restoring moment divided by overturning moment about the currently governing edge.

xR=∑MC∑V,∣e∣≤B/6;(full contact),∣e∣=B/2;(tipping)x_R=\frac{\sum M_C}{\sum V},\quad |e|\le B/6\\;\text{(full contact)},\quad |e|=B/2\\;\text{(tipping)}

Equation concept

For B/6 < |e| < B/2, the model uses compression-only partial contact. A resultant beyond B/2 is not drawn as a physical reaction because the assumed base equilibrium has been lost.

Simulation 2 Concept Question

At what load position does the resultant first leave the middle third?

Overturning Safety-Factor Form

Pedagogical ratio of restoring to overturning moment.

FSOT=∑Mrestoring∑MoverturningFS_{\mathrm{OT}}=\frac{\sum M_{\mathrm{restoring}}}{\sum M_{\mathrm{overturning}}}

Variables

SymbolDescriptionUnit
FSOTFS_{\mathrm{OT}}Idealized factor against overturningunitless

Simulation 3 Instructions

Use the retaining-wall scenario to compare the simplified lateral resultant acting at H/3H/3 with wall-weight restoring moment, sliding resistance, and the compression-only base reaction.

Stability and Base-Resultant Suite

Concept and model scope

Apply a lateral resultant at H/3 and compare sliding, restoring/overturning moments, and compression-only base contact.

Simulation purpose: Move real loads and geometry while the signed moment balance, compression-only contact state, and reaction line update together.

Model scope: Rigid-base statics model with compression-only support contact. Middle-third exit indicates partial contact; physical base-edge exit is a distinct loss-of-equilibrium limit.

Verification: The reaction line is drawn only while it lies on the physical base. Signed load offsets, H/3 for the retaining-wall resultant, and the displayed compression zone use the same model coordinates as the calculations.

Centered weight

Centered weight

Centered downward weight through the base center. It contributes stabilizing moment about either edge.

300 kN
Base width

Base width

Physical support width B. Kern boundaries are ±B/6 and the physical edges are ±B/2.

3.0 m
Lateral resultant

Lateral resultant

Simplified lateral resultant magnitude. Its line of action is fixed at H/3 above the base.

60 kN
Wall height

Wall height

Physical wall height H. The simplified lateral resultant is shown and evaluated at H/3.

2.0 m
Static friction coefficient

Static friction coefficient

Defines the limiting base sliding resistance μsΣV. The actual static friction is only the horizontal demand while equilibrium remains admissible.

0.45
RW 300 kNH 60 kNFbaseB = 3 mkern ±0.50 m · edges ±1.50 m
full compression contact
Reaction location xR
+0.133 m

Reaction location xR

Signed from the base center. Positive is toward the positive overturning edge.

Eccentricity |e|
0.133 m

Eccentricity |e|

Kern = 0.500 m; edge = 1.500 m.

Compression width
3.000 m

Compression width

Full base width is active inside the kern. After kern exit, a compression-only triangular contact zone is used; no tensile base pressure is reported.

Contact state
full-contact

Contact state

Kern exit and physical base-edge exit are intentionally separate states.

Sliding state
static

Sliding state

Static friction supplies the demand only while demand ≤ μsΣV.

Sliding factor
2.250

Sliding factor

Limiting friction capacity divided by horizontal demand.

Stabilizing moment
450 kN·m
Overturning moment
40 kN·m
Overturning factor
11.250

Overturning factor

Restoring moment divided by overturning moment about the currently governing edge.

xR=∑MC∑V,∣e∣≤B/6;(full contact),∣e∣=B/2;(tipping)x_R=\frac{\sum M_C}{\sum V},\quad |e|\le B/6\\;\text{(full contact)},\quad |e|=B/2\\;\text{(tipping)}

Equation concept

For B/6 < |e| < B/2, the model uses compression-only partial contact. A resultant beyond B/2 is not drawn as a physical reaction because the assumed base equilibrium has been lost.

Simulation 3 Concept Question

Why is a retaining wall with a sufficient overturning moment ratio not automatically a compliant design?

Simulation 4 Instructions

Adjust crane load, positive lift outreach, counterweight, negative counterweight offset, and base width. The signed horizontal offsets must shift the base resultant according to the actual moment balance.

Stability and Base-Resultant Suite

Concept and model scope

Use signed lifted-load and counterweight offsets so the reaction line follows the actual moment balance.

Simulation purpose: Move real loads and geometry while the signed moment balance, compression-only contact state, and reaction line update together.

Model scope: Rigid-base statics model with compression-only support contact. Middle-third exit indicates partial contact; physical base-edge exit is a distinct loss-of-equilibrium limit.

Verification: The reaction line is drawn only while it lies on the physical base. Signed load offsets, H/3 for the retaining-wall resultant, and the displayed compression zone use the same model coordinates as the calculations.

Centered weight

Centered weight

Centered downward weight through the base center. It contributes stabilizing moment about either edge.

300 kN
Base width

Base width

Physical support width B. Kern boundaries are ±B/6 and the physical edges are ±B/2.

3.0 m
Lifted load

Lifted load

Downward lifted load on the positive-outreach side of the crane.

60 kN
Lift outreach

Lift outreach

Positive signed horizontal offset of the lifted load from the base center.

2.0 m
Counterweight

Counterweight

Downward counterweight on the negative side of the base center.

100 kN
Counterweight radius

Counterweight radius

Magnitude of the counterweight lever arm. Its signed offset is negative in the moment balance.

2.5 m
RW 300 kNLift 60 kNCounter 100 kNB = 3 mkern ±0.50 m · edges ±1.50 m
full compression contact
Reaction location xR
-0.283 m

Reaction location xR

Signed from the base center. Positive is toward the positive overturning edge.

Eccentricity |e|
0.283 m

Eccentricity |e|

Kern = 0.500 m; edge = 1.500 m.

Compression width
3.000 m

Compression width

Full base width is active inside the kern. After kern exit, a compression-only triangular contact zone is used; no tensile base pressure is reported.

Contact state
full-contact

Contact state

Kern exit and physical base-edge exit are intentionally separate states.

Sliding state
not-applicable

Sliding state

No horizontal sliding demand exists in this scenario.

Sliding factor
Not applicable

Sliding factor

Limiting friction capacity divided by horizontal demand.

Stabilizing moment
660 kN·m
Overturning moment
100 kN·m
Overturning factor
6.600

Overturning factor

Restoring moment divided by overturning moment about the currently governing edge.

xR=∑MC∑V,∣e∣≤B/6;(full contact),∣e∣=B/2;(tipping)x_R=\frac{\sum M_C}{\sum V},\quad |e|\le B/6\\;\text{(full contact)},\quad |e|=B/2\\;\text{(tipping)}

Equation concept

For B/6 < |e| < B/2, the model uses compression-only partial contact. A resultant beyond B/2 is not drawn as a physical reaction because the assumed base equilibrium has been lost.

Simulation 4 Concept Question

Which has greater influence on tipping resistance: counterweight magnitude or counterweight lever arm?

Simulation 5 Instructions

Move the resultant through the base and distinguish full contact, middle-third exit with partial compression contact, the physical base edge, and complete loss of the assumed base reaction.

Stability and Base-Resultant Suite

Concept and model scope

Separate middle-third exit from physical base-edge exit and show the compression-only contact width.

Simulation purpose: Move real loads and geometry while the signed moment balance, compression-only contact state, and reaction line update together.

Model scope: Rigid-base statics model with compression-only support contact. Middle-third exit indicates partial contact; physical base-edge exit is a distinct loss-of-equilibrium limit.

Verification: The reaction line is drawn only while it lies on the physical base. Signed load offsets, H/3 for the retaining-wall resultant, and the displayed compression zone use the same model coordinates as the calculations.

Centered weight

Centered weight

Centered downward weight through the base center. It contributes stabilizing moment about either edge.

300 kN
Base width

Base width

Physical support width B. Kern boundaries are ±B/6 and the physical edges are ±B/2.

3.0 m
Horizontal load

Horizontal load

Horizontal force that creates a positive overturning moment and shifts the base resultant toward the positive edge.

60 kN
Load height

Load height

Vertical distance from the base to the horizontal force line of action.

2.0 m
RW 300 kNH 60 kNFbaseB = 3 mkern ±0.50 m · edges ±1.50 m
full compression contact
Reaction location xR
+0.400 m

Reaction location xR

Signed from the base center. Positive is toward the positive overturning edge.

Eccentricity |e|
0.400 m

Eccentricity |e|

Kern = 0.500 m; edge = 1.500 m.

Compression width
3.000 m

Compression width

Full base width is active inside the kern. After kern exit, a compression-only triangular contact zone is used; no tensile base pressure is reported.

Contact state
full-contact

Contact state

Kern exit and physical base-edge exit are intentionally separate states.

Sliding state
static

Sliding state

Static friction supplies the demand only while demand ≤ μsΣV.

Sliding factor
2.250

Sliding factor

Limiting friction capacity divided by horizontal demand.

Stabilizing moment
450 kN·m
Overturning moment
120 kN·m
Overturning factor
3.750

Overturning factor

Restoring moment divided by overturning moment about the currently governing edge.

xR=∑MC∑V,∣e∣≤B/6;(full contact),∣e∣=B/2;(tipping)x_R=\frac{\sum M_C}{\sum V},\quad |e|\le B/6\\;\text{(full contact)},\quad |e|=B/2\\;\text{(tipping)}

Equation concept

For B/6 < |e| < B/2, the model uses compression-only partial contact. A resultant beyond B/2 is not drawn as a physical reaction because the assumed base equilibrium has been lost.

Simulation 5 Concept Question

Why can the resultant leave the middle third before the body reaches complete overturning?

Stability Analysis Procedure

  1. Draw the free-body diagram and identify a possible tipping edge.
  2. Sum vertical forces to obtain the total compressive reaction.
  3. Sum moments to locate the base resultant.
  4. Compare eccentricity with B/6B/6 and B/2B/2.
  5. Compute sliding and tipping thresholds separately.
  6. Reject any assumed contact state that requires tension.
  7. Report the governing equilibrium mode without extending the result into code compliance.
Rigid-Body Stability Limit-State Workflow

Evaluate compressive contact, kern or contact-footprint limits, incipient tipping, and sliding as distinct rigid-body equilibrium states without applying the rectangular-base B/6 rule outside its assumptions.

Rigid-Body Stability Limit-State WorkflowEvaluate compressive contact, kern or contact-footprint limits, incipient tipping, and sliding as distinct rigid-body equilibrium states without applying the rectangular-base B/6 rule outside its assumptions.. Define the rigid body, load path, base geometry, and candidate tipping edges → For a varying load, compare all equilibrium thresholds along the same loading path; For a varying load, compare all equilibrium thresholds along the same loading path → Compute the compressive normal resultant and base-reaction location from equilibrium; Compute the compressive normal resultant and base-reaction location from equilibrium → Total normal resultant is compressive (V > 0)?; Total normal resultant is compressive (V > 0)? — No → Assumed compressive contact state is impossible; Total normal resultant is compressive (V > 0)? — Yes → Resultant lies within the base contact footprint?; Resultant lies within the base contact footprint? — No → Resultant beyond the edge: overturning / loss of equilibrium; Resultant lies within the base contact footprint? — Yes → Resultant is located at the tipping edge within tolerance?; Resultant is located at the tipping edge within tolerance? — Yes → Incipient tipping: overturning threshold reached at base edge; Resultant is located at the tipping edge within tolerance? — No → Resultant lies within the compression kern (e.g. |e| ≤ B/6 for rectangular bases)?; Incipient tipping: overturning threshold reached at base edge → Report the governing equilibrium mode; Resultant lies within the compression kern (e.g. |e| ≤ B/6 for rectangular bases)? — Yes → Full-base compression: entire contact surface is in compression; Resultant lies within the compression kern (e.g. |e| ≤ B/6 for rectangular bases)? — No → Partial contact or uplift: recompute the admissible contact zone; Full-base compression: entire contact surface is in compression → Sliding resistance is adequate for the current contact state?; Partial contact or uplift: recompute the admissible contact zone → Sliding resistance is adequate for the current contact state?; Sliding resistance is adequate for the current contact state? — Yes → Report the governing equilibrium mode; Sliding resistance is adequate for the current contact state? — No → Sliding is the active or earlier limit state; Sliding is the active or earlier limit state → Report the governing equilibrium mode

Define the rigid body, load path, base geometry, and candidate tipping edges → For a varying load, compare all equilibrium thresholds along the same loading path; For a varying load, compare all equilibrium thresholds along the same loading path → Compute the compressive normal resultant and base-reaction location from equilibrium; Compute the compressive normal resultant and base-reaction location from equilibrium → Total normal resultant is compressive (V > 0)?; Total normal resultant is compressive (V > 0)? — No → Assumed compressive contact state is impossible; Total normal resultant is compressive (V > 0)? — Yes → Resultant lies within the base contact footprint?; Resultant lies within the base contact footprint? — No → Resultant beyond the edge: overturning / loss of equilibrium; Resultant lies within the base contact footprint? — Yes → Resultant is located at the tipping edge within tolerance?; Resultant is located at the tipping edge within tolerance? — Yes → Incipient tipping: overturning threshold reached at base edge; Resultant is located at the tipping edge within tolerance? — No → Resultant lies within the compression kern (e.g. |e| ≤ B/6 for rectangular bases)?; Incipient tipping: overturning threshold reached at base edge → Report the governing equilibrium mode; Resultant lies within the compression kern (e.g. |e| ≤ B/6 for rectangular bases)? — Yes → Full-base compression: entire contact surface is in compression; Resultant lies within the compression kern (e.g. |e| ≤ B/6 for rectangular bases)? — No → Partial contact or uplift: recompute the admissible contact zone; Full-base compression: entire contact surface is in compression → Sliding resistance is adequate for the current contact state?; Partial contact or uplift: recompute the admissible contact zone → Sliding resistance is adequate for the current contact state?; Sliding resistance is adequate for the current contact state? — Yes → Report the governing equilibrium mode; Sliding resistance is adequate for the current contact state? — No → Sliding is the active or earlier limit state; Sliding is the active or earlier limit state → Report the governing equilibrium mode

  • Define the rigid body, load path, base geometry, and candidate tipping edges: terminator
  • For a varying load, compare all equilibrium thresholds along the same loading path: process
  • Compute the compressive normal resultant and base-reaction location from equilibrium: process
  • Total normal resultant is compressive (V > 0)?: decision
  • Assumed compressive contact state is impossible: terminator
  • Resultant lies within the base contact footprint?: decision
  • Resultant beyond the edge: overturning / loss of equilibrium: terminator
  • Resultant is located at the tipping edge within tolerance?: decision
  • Incipient tipping: overturning threshold reached at base edge: process
  • Resultant lies within the compression kern (e.g. |e| ≤ B/6 for rectangular bases)?: decision
  • Full-base compression: entire contact surface is in compression: process
  • Partial contact or uplift: recompute the admissible contact zone: process
  • Sliding resistance is adequate for the current contact state?: decision
  • Sliding is the active or earlier limit state: process
  • Report the governing equilibrium mode: terminator
Key Takeaways
  • The base reaction shifts to satisfy moment equilibrium.
  • The middle third is a full-compression region for a linear rectangular-base model.
  • Leaving the kern is different from complete tipping.
  • Sliding, tipping, and uplift are distinct limiting states.
  • Stability equilibrium alone is not a complete structural or geotechnical design check.